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Question #43684

Difficulty: 8/10 | 🟢 Free

The Question

A radioactive sample initially contains $N_0$ nuclei of isotope $X$ . It undergoes a successive decay chain $X \xrightarrow{\lambda_1} Y \xrightarrow{\lambda_2} Z$ , where $Z$ is stable. The total mass defect for the complete transformation of one nucleus of $X$ to $Z$ is $\Delta m$ .

To model the population of $Y$ , a physicist constructs a source-free series L-C-R circuit. The charge $q(t)$ on the capacitor is scaled such that $q(t) = \alpha N_Y(t)$ , where $\alpha$ is a constant with units of Coulombs. At $t=0$ , the capacitor is uncharged. The circuit parameters $R$ , $L$ , and $C$ are tuned such that the differential equation governing $q(t)$ perfectly matches the second-order differential equation governing $N_Y(t)$ for all $t \ge 0$ .

If the total electrical energy dissipated as heat in the resistor over $t \in [0, \infty)$ is exactly equal to the total rest mass energy released by the complete decay of the initial sample, which of the following expressions gives the required resistance $R$ ? (Here $c$ is the speed of light in vacuum).

Options

A.
$\frac{2 \Delta m c^2 (\lambda_1 + \lambda_2)}{\alpha^2 \lambda_1^2 N_0}$
B.
$\frac{\Delta m c^2 (\lambda_1 + \lambda_2)}{\alpha^2 \lambda_1^2 N_0}$
C.
$\frac{2 \Delta m c^2 \lambda_2}{\alpha^2 \lambda_1^2 N_0}$
D.
$\frac{2 \Delta m c^2 (\lambda_1 + \lambda_2)}{\alpha^2 N_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Mass Defect vs. Binding Energy|Successive Decay Equilibrium|L-C Circuit Analogy

Why This Question Is Tricky

Common Mistakes

This question tests the structural analogy between coupled first-order decay processes and second-order damped oscillations, combined with mass-energy equivalence.

A common trap is attempting to equate the instantaneous electrical power ( $i^2 R$ ) to the instantaneous nuclear power release, which breaks down because the analogy holds at the level of the characteristic equation and global energy boundary conditions, not local power.

Furthermore, students often incorrectly assume the damping term $R/L$ maps solely to the decay constant of the daughter nucleus ( $\lambda_2$ ), failing to derive the full second-order Bateman differential equation which reveals that the effective damping is the sum of both decay constants ( $\lambda_1 + \lambda_2$ ).

Finally, calculating the initial inductive energy requires recognizing that the initial rate of change of the daughter nucleus population dictates the initial current.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Mass Defect vs. Binding Energy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free